Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations

The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on invariant subspace variations that are proportional to the reciprocals of absolute gaps between subsets of spectra or subsets of singular values. These bounds may be bad news for invariant subspaces corresponding to clustered eigenvalues or clustered singular values of much smaller magnitudes than the norms of matrices under considerations when some of these clustered eigenvalues or clustered singular values are perfectly relatively distinguishable from the rest. In this paper, we consider how eigenspaces of a Hermitian matrix A change when it is perturbed to e A = D AD and how singular values of a (nonsquare) matrix B change when it is perturbed to e B = D 1 BD2, where D, D1 and D2 are assumed to be close to identity matrices of suitable dimensions, or either D1 or D2 close to some unitary matrix. It is proved that under these kinds of perturbations, the change of invariant subspaces are proportional to the reciprocals of relative gaps between subsets of spectra or subsets of singular values. We have been able to extend well-known Davis-Kahan sin theorems and Wedin sin theorems. As applications, we obtained bounds for perturbations of graded matrices. This material is based in part upon work supported, during January, 1992{August, 1995, by Argonne National Laboratory under grant No. 20552402 and by the University of Tennessee through the Advanced Research Projects Agency under contract No. DAAL03-91-C-0047, by the National Science Foundation under grant No. ASC-9005933, and by the National Science Infrastructure grants No. CDA-8722788 and CDA-9401156, and supported, since August, 1995, by a Householder Fellowship in Scienti c Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, O ce of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp. Part of this work was done during summer of 1994 while the author was at Department of Mathematics, University of California at Berkeley.